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A condenser is simply a capacitor which prevents voltage from arcing across the points when they open, in turn preventing premature wear on the points.

The condenser prevents arcing as follows. The condenser is wired in parallel with the points. When the points open, the voltage across the condenser can not change instantaneously. Because of this, the voltage will slowly rise across the capacitor as it charges. Because the points are wired in parallel with the capacitor, the voltage across the point gap will also rise slowly, preventing an arc.

The condenser on a Tomos A3. On this moped (and on many others) the condenser is located on the stator plate, and is accessed by removing the flywheel.

There is an easier way to think of the condenser/capacitor's operation: As the points just start to open the flow of current will start to jump the gap, but as this is happening the resistance in the points circuit is also going up. The capacitor is at zero resistance at this instant so the current will flow into the condenser rather than jump the gap.(current always takes the path of least resistance) As this is happening the points will get so far open there is not enough voltage in the system to jump the gap, so no arching takes place. When the points close the capacitor drains it current back the ground thus setting its resistance back to zero, making it ready for the next cycle.

If the condenser is shorted internally it won't matter that the points are opening and closing because the current will always be flowing to ground through the condenser. If the current doesn't go on and off through the coil you get no spark.

To test the condseor you have to completely isolate it from the circuit and test it with an ohm meter. If it is shorted it will show zero ohms. If it is good it will show a range of ohms as it slowly fills then ends at open circuit, infinite ohms. Use a jumper from the center wire to its case and it will discharge and you can repeat the test. Heat and the high voltage produced at high speed will eventually kill the condenser. They usually die the way a light bulb dies, all at once the next time you surge voltage through it when you try you start the engine.